Looking for generalization of projective model structure

Question

If $\mathcal{M}$ is a cofibrantly generated model category and $\mathcal{C}$ is a small category, then we can give $\mathcal{M}^\mathcal{C}$ the projective model structure, in which weak equivalences and fibrations are transformations that are such on each object of $\mathcal{C}$. I recall seeing a generalization in which we take a subset $D$ of objects of $\mathcal{C}$ and say that a transformation $f$ is a weak equivalence or fibration if $f(d)$ is one for every $d\in D$ (and require nothing at the other objects). Am I remembering correctly that this works in this generality, and does anyone have a reference for where this is written down? I'd like to use this but see no need to reinvent the wheel if I can just refer to it.

Answer 1

This model structure has appeared explicitly in the paper by Paul Balmer and Michel Matthey "Codescent theory I: Foundations". Theorem 3.5 establishes the existence of, what the authors called, the relative model structure.

But the ideas behind this model structure go back to Dwyer-Kan concept of an orbit introduced in the paper "Singular functors and realization functors". If you choose the set of orbits to be the representable functors $hom_{\mathcal{C}}(d,-)$ for all $d\in \mathcal{D}$, then this set of orbits defines the same $\mathcal{D}$-relative model structure on the category of $\mathcal{C}$-indexed diagrams.

Answer 2

We can apply the Kan transfer theorem (Theorem 11.3.2 in Hirschhorn) to the right adjoint functor U: M^C→M^D, where M^D is equipped with the projective model structure. Its left adjoint F sends representable functors y(X) on D to the corresponding representable functors y(X) on C. The conditions in the theorem boil down to verifying that U takes relative F(y(X)⊗J)-complexes to weak equivalences (J denotes a set of generating acyclic cofibrations of M), where X∈D.

However, F(y_D(X)⊗J)=y_C(X)⊗J is a subclass of projective acylic cofibrations (which are generated by y_C(X)⊗J for all X∈C), and every projective acyclic cofibration is an objectwise weak equivalence, which completes the proof.

Answer 3

You might be thinking of the paper "On Modified Reedy and Modified Projective Model Structures" by Mark Johnson. It does exactly what you asked in your question, in Proposition 6.4. Funny you should mention reinventing the wheel; I reproved this result a few years ago in a mathoverflow answer, before I knew about Mark's paper.